Abstract tropical linear programming

TL;DR

This paper introduces a combinatorial framework for tropical linear programming, extending classical methods and connecting to mean payoff games and scheduling through polyhedral and graph-based structures.

Contribution

It develops a novel combinatorial abstraction called signed tropical matroids, generalizing feasible point search in tropical linear systems and linking to classical optimization and game theory.

Findings

Introduces signed tropical matroids as a new combinatorial structure.

Establishes connections between tropical linear programming, classical simplex, and mean payoff games.

Provides polyhedral and graph-based insights into tropical optimization problems.

Abstract

In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the product of two simplices and the combinatorics of the associated set of bipartite graphs with an additional sign information which we call a signed tropical matroid. We demonstrate the connections with the classical simplex method, mean payoff games and scheduling.